- FindStat

Identifier

St000810: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        []=>1
[1]=>1
[2]=>1
[1,1]=>3
[3]=>1
[2,1]=>2
[1,1,1]=>10
[4]=>1
[3,1]=>2
[2,2]=>3
[2,1,1]=>7
[1,1,1,1]=>47
[5]=>1
[4,1]=>2
[3,2]=>2
[3,1,1]=>6
[2,2,1]=>6
[2,1,1,1]=>26
[1,1,1,1,1]=>246
[6]=>1
[5,1]=>2
[4,2]=>2
[4,1,1]=>6
[3,3]=>3
[3,2,1]=>6
[3,1,1,1]=>24
[2,2,2]=>10
[2,2,1,1]=>26
[2,1,1,1,1]=>138
[1,1,1,1,1,1]=>1602
[7]=>1
[6,1]=>2
[5,2]=>2
[5,1,1]=>6
[4,3]=>2
[4,2,1]=>5
[4,1,1,1]=>23
[3,3,1]=>6
[3,2,2]=>6
[3,2,1,1]=>20
[3,1,1,1,1]=>114
[2,2,2,1]=>23
[2,2,1,1,1]=>105
[2,1,1,1,1,1]=>767
[1,1,1,1,1,1,1]=>11481
[8]=>1
[7,1]=>2
[6,2]=>2
[6,1,1]=>6
[5,3]=>2
[5,2,1]=>5
[5,1,1,1]=>23
[4,4]=>3
[4,3,1]=>6
[4,2,2]=>7
[4,2,1,1]=>19
[4,1,1,1,1]=>111
[3,3,2]=>6
[3,3,1,1]=>22
[3,2,2,1]=>20
[3,2,1,1,1]=>90
[3,1,1,1,1,1]=>652
[2,2,2,2]=>47
[2,2,2,1,1]=>111
[2,2,1,1,1,1]=>599
[2,1,1,1,1,1,1]=>5295
[1,1,1,1,1,1,1,1]=>95503
[9]=>1
[8,1]=>2
[7,2]=>2
[7,1,1]=>6
[6,3]=>2
[6,2,1]=>5
[6,1,1,1]=>23
[5,4]=>2
[5,3,1]=>5
[5,2,2]=>6
[5,2,1,1]=>18
[5,1,1,1,1]=>110
[4,4,1]=>6
[4,3,2]=>5
[4,3,1,1]=>19
[4,2,2,1]=>18
[4,2,1,1,1]=>80
[4,1,1,1,1,1]=>622
[3,3,3]=>10
[3,3,2,1]=>21
[3,3,1,1,1]=>91
[3,2,2,2]=>23
[3,2,2,1,1]=>85
[3,2,1,1,1,1]=>471
[3,1,1,1,1,1,1]=>4285
[2,2,2,2,1]=>110
[2,2,2,1,1,1]=>512
[2,2,1,1,1,1,1]=>3586
[2,1,1,1,1,1,1,1]=>39468
[1,1,1,1,1,1,1,1,1]=>871030
[10]=>1
[9,1]=>2
[8,2]=>2
[8,1,1]=>6
[7,3]=>2
[7,2,1]=>5
[7,1,1,1]=>23
[6,4]=>2
[6,3,1]=>5
[6,2,2]=>6
[6,2,1,1]=>18
[6,1,1,1,1]=>110
[5,5]=>3
[5,4,1]=>6
[5,3,2]=>6
[5,3,1,1]=>18
[5,2,2,1]=>18
[5,2,1,1,1]=>78
[5,1,1,1,1,1]=>618
[4,4,2]=>6
[4,4,1,1]=>22
[4,3,3]=>6
[4,3,2,1]=>18
[4,3,1,1,1]=>84
[4,2,2,2]=>26
[4,2,2,1,1]=>78
[4,2,1,1,1,1]=>426
[4,1,1,1,1,1,1]=>4086
[3,3,3,1]=>23
[3,3,2,2]=>22
[3,3,2,1,1]=>82
[3,3,1,1,1,1]=>470
[3,2,2,2,1]=>86
[3,2,2,1,1,1]=>412
[3,2,1,1,1,1,1]=>2866
[3,1,1,1,1,1,1,1]=>32048
[2,2,2,2,2]=>246
[2,2,2,2,1,1]=>582
[2,2,2,1,1,1,1]=>3134
[2,2,1,1,1,1,1,1]=>26038
[2,1,1,1,1,1,1,1,1]=>340198
[1,1,1,1,1,1,1,1,1,1]=>8879558
[11]=>1
[10,1]=>2
[9,2]=>2
[9,1,1]=>6
[8,3]=>2
[8,2,1]=>5
[8,1,1,1]=>23
[7,4]=>2
[7,3,1]=>5
[7,2,2]=>6
[7,2,1,1]=>18
[7,1,1,1,1]=>110
[6,5]=>2
[6,4,1]=>5
[6,3,2]=>5
[6,3,1,1]=>17
[6,2,2,1]=>17
[6,2,1,1,1]=>77
[6,1,1,1,1,1]=>617
[5,5,1]=>6
[5,4,2]=>5
[5,4,1,1]=>19
[5,3,3]=>6
[5,3,2,1]=>17
[5,3,1,1,1]=>75
[5,2,2,2]=>23
[5,2,2,1,1]=>73
[5,2,1,1,1,1]=>407
[5,1,1,1,1,1,1]=>4041
[4,4,3]=>6
[4,4,2,1]=>17
[4,4,1,1,1]=>87
[4,3,3,1]=>19
[4,3,2,2]=>18
[4,3,2,1,1]=>70
[4,3,1,1,1,1]=>418
[4,2,2,2,1]=>77
[4,2,2,1,1,1]=>363
[4,2,1,1,1,1,1]=>2545
[4,1,1,1,1,1,1,1]=>30319
[3,3,3,2]=>23
[3,3,3,1,1]=>87
[3,3,2,2,1]=>79
[3,3,2,1,1,1]=>383
[3,3,1,1,1,1,1]=>2751
[3,2,2,2,2]=>110
[3,2,2,2,1,1]=>398
[3,2,2,1,1,1,1]=>2326
[3,2,1,1,1,1,1,1]=>19742
[3,1,1,1,1,1,1,1,1]=>268350
[2,2,2,2,2,1]=>617
[2,2,2,2,1,1,1]=>2855
[2,2,2,1,1,1,1,1]=>19853
[2,2,1,1,1,1,1,1,1]=>201403
[2,1,1,1,1,1,1,1,1,1]=>3166097
[1,1,1,1,1,1,1,1,1,1,1]=>98329551
[12]=>1
[11,1]=>2
[10,2]=>2
[10,1,1]=>6
[9,3]=>2
[9,2,1]=>5
[9,1,1,1]=>23
[8,4]=>2
[8,3,1]=>5
[8,2,2]=>6
[8,2,1,1]=>18
[8,1,1,1,1]=>110
[7,5]=>2
[7,4,1]=>5
[7,3,2]=>5
[7,3,1,1]=>17
[7,2,2,1]=>17
[7,2,1,1,1]=>77
[7,1,1,1,1,1]=>617
[6,6]=>3
[6,5,1]=>6
[6,4,2]=>6
[6,4,1,1]=>18
[6,3,3]=>7
[6,3,2,1]=>17
[6,3,1,1,1]=>73
[6,2,2,2]=>24
[6,2,2,1,1]=>72
[6,2,1,1,1,1]=>404
[6,1,1,1,1,1,1]=>4036
[5,5,2]=>6
[5,5,1,1]=>22
[5,4,3]=>5
[5,4,2,1]=>17
[5,4,1,1,1]=>83
[5,3,3,1]=>18
[5,3,2,2]=>19
[5,3,2,1,1]=>69
[5,3,1,1,1,1]=>387
[5,2,2,2,1]=>75
[5,2,2,1,1,1]=>349
[5,2,1,1,1,1,1]=>2451
[5,1,1,1,1,1,1,1]=>29997
[4,4,4]=>10
[4,4,3,1]=>21
[4,4,2,2]=>26
[4,4,2,1,1]=>74
[4,4,1,1,1,1]=>450
[4,3,3,2]=>18
[4,3,3,1,1]=>78
[4,3,2,2,1]=>73
[4,3,2,1,1,1]=>345
[4,3,1,1,1,1,1]=>2505
[4,2,2,2,2]=>138
[4,2,2,2,1,1]=>378
[4,2,2,1,1,1,1]=>2098
[4,2,1,1,1,1,1,1]=>17682
[4,1,1,1,1,1,1,1,1]=>253818
[3,3,3,3]=>47
[3,3,3,2,1]=>95
[3,3,3,1,1,1]=>419
[3,3,2,2,2]=>88
[3,3,2,2,1,1]=>392
[3,3,2,1,1,1,1]=>2204
[3,3,1,1,1,1,1,1]=>18908
[3,2,2,2,2,1]=>437
[3,2,2,2,1,1,1]=>2233
[3,2,2,1,1,1,1,1]=>15389
[3,2,1,1,1,1,1,1,1]=>154649
[3,1,1,1,1,1,1,1,1,1]=>2501957
[2,2,2,2,2,2]=>1602
[2,2,2,2,2,1,1]=>3546
[2,2,2,2,1,1,1,1]=>19274
[2,2,2,1,1,1,1,1,1]=>154210
[2,2,1,1,1,1,1,1,1,1]=>1807634
[2,1,1,1,1,1,1,1,1,1,1]=>33022346
[1,1,1,1,1,1,1,1,1,1,1,1]=>1191578522
                    

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Description

The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions.
For example, $p_{22} = 2m_{22} + m_4$, so the statistic on the partition $22$ is 3.

Code

def statistic(mu):
    m = SymmetricFunctions(ZZ).m()
    p = SymmetricFunctions(ZZ).p()
    return sum(coeff for _, coeff in m(p(mu)))

Created

May 20, 2017 at 17:57 by Martin Rubey

Updated

Oct 31, 2017 at 08:07 by Martin Rubey